When the interaction among the players is through the state only, a rigorous connection between the \(N\) -player and the mean field game was first established by [1]} and [2]}, proving compactness results for sequences of Nash equilibria, using relaxed controls.
[3]} used existence and regularity properties of solutions of the so-called master equation, a PDE on the space of probability measures characterizing the mean field game to prove convergence results for the value functions.
Moreover, under proper smoothness conditions on the solution (when it exists) of the master equation, [4]}, [5]} further analyzed the mean field game limit for the state processes at equilibrium of the game with common noise.
Notably, [4]} derive the large deviation principle for closed-loop equilibria in the uniqueness regime (i.e. when the limiting mean field game admits a unique mean field equilibrium).
